The American military presence in Pakistan has already grown substantially over the past year, and now totals more than two hundred troops, part of a program to share intelligence with Pakistani Army and paramilitary troops and train them to battle militant groups.

But the failed bombing in Times Square, and evidence that the accused man, Faisal Shahzad, received training in a camp run by the Pakistani Taliban, has given support to those who want to expand the mission.

In particular, some inside the administration believe that the C.I.A. program of killing militants from the air is insufficient for preventing attacks on the West, and that an expanded training mission might raise confidence in Pakistan’s military enough to launch an offensive in the militant sanctuary of North Waziristan, in the tribal areas.

Officials, who requested anonymity to discuss the strategy surrounding the secretive program, emphasized that any new troops in Pakistan would serve as advisers and trainers, not as combat forces.

But the presence of any American troops on Pakistani soil is extremely sensitive. It is thought to be widely opposed by Pakistanis, and the Pentagon has worked hard to keep a low profile. American troops there are careful about how much time they spend away from enclosed garrisons.

Officials said there was now discussion about presenting Pakistan’s government with a formal request to dispatch more Special Operations troops to the country. American officials believe they have improved relations with Islamabad in recent months, and that this might be a particularly opportune time to press the case.

But one senior Pakistani official cautioned that Washington should not overreach.

“The Americans have to be careful not to make demands that are disproportionate to the good will they have built up,” he said.

It is also unclear how much leverage the United States would have, given that the attack was amateurish and unsuccessful.

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In the meantime, American officials said that Pakistan’s government had been helpful during the initial phase of the investigation. Investigators have passed along leads from Mr. Shahzad’s interrogation to officials at the American Embassy in Islamabad, who have passed the leads on to Pakistani authorities.

So far, administration officials said, Pakistani authorities have been cooperating with requests for details about Mr. Shahzad and his family. The American ambassador to Pakistan, Anne Patterson, spoke on Thursday with Pakistan’s prime minister, Yousaf Raza Gilani, and the foreign minister, Shah Mehmood Qureshi.

Administration officials said their top priority was to nail down Mr. Shahzad’s links to militant groups, and then to press Pakistan to act against the groups. While the evidence continues to point to the Pakistani Taliban as the primary link, a senior official said Mr. Shahzad “appears to be at the intersection of a whole lot of strands.”

“There’s a bit of a false distinction being made between these groups,” said another official. “The Pakistani Taliban is connected to Al Qaeda, which is connected to the Haqqani network. I don’t think you can put team jerseys on them.”

Pakistani officials have blown hot and cold on the issue of American troops in the country. Months ago, when sentiment was running more strongly against additional troops, Pakistan held up issuing visas for advisers and trainers. After visits by senior officials, including Adm. Mike Mullen, the chairman of the Joint Chiefs of Staff, Pakistan began issuing them again.

The near-miss in Times Square on Saturday evening is likely to make some Pakistani officials less reluctant to accept additional American trainers, said officials with knowledge of the Pakistani government. There is a sense in Islamabad, these officials said, that if the car bomb had exploded, it would have severely strained relations between the United States and Pakistan.

Gen. David H. Petraeus, the head of United States Central Command, has been very supportive of the American training mission in Pakistan. But military officials said General Petraeus was cautious about making a formal request to Pakistan now, as he is concerned about the impact such a move would have on relations with Pakistan’s military and on inflaming anti-American sentiment in the country.

For the Obama administration, the terrorist plot comes at a sensitive time in its effort to cultivate Pakistan. In March, it held a high-level strategic dialogue with Pakistan’s government, which officials said went a long way toward building up trust between the two sides.

Pakistan, for its part, said it would crack down on any group making the United States a target. “We do not, and we will not, make distinctions between any terrorist groups,” said Pakistan’s ambassador to the United States, Husain Haqqani. “Pakistan has proven over the last few years that it is fighting extremists for its own sake. We will continue to do that.”

Eric Schmitt contributed reporting.

A version of this article appears in print on May 7, 2010, on Page A12 of the New York edition with the headline: Debate on Expanded Presence in Pakistan. Order Reprints|Today's Paper|Subscribe